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Composite Matrix Transformation - Reflection

  1. Nov 24, 2008 #1
    1. The problem statement, all variables and given/known data

    Let T1 be the reflection about the line 2x–5y=0 and T2 be the reflection about the line –4x+3y=0 in the euclidean plane.
    (i) The standard matrix of T1 o T2 is: ?

    Thus T1 o T2 is a counterclockwise rotation about the origin by an angle of _ radians?

    (ii) The standard matrix of T2 o T1 is: ?

    Thus T2 o T1 is a counterclockwise rotation about the origin by an angle of _ radians?

    2. Relevant equations

    I think these equations are correct...

    T(v) = A(v)

    Reflection:
    A =
    [((2(u_1))^2)), (2(u_1)(u_2)))
    (2(u_1)(u_2)), ((2(u_2))^2))]
    *u being the unit vectors

    Rotation counterclockwise:
    A =
    [cosx -sinx
    sinx cosx]

    S o T is the matrix Transformation with matrix AB

    3. The attempt at a solution

    I thought I understood this, but again, I guess I've understood something incorrectly.

    For the first question, I got the unit vectors to be:
    [(5/sqrt29)], (2/sqrt29)] and [(3/5), (4/5)] for T_1 and T_2 respectively.
    I then got the standard matrix A of T_1 to be:
    [(21/29) (20/29)
    (20/29) (-21/29)]
    and the standard matrix B of T_2 to be:
    [(-7/25) (24/25)
    (24/25) (7/25)]

    I then took AB = the dot product of these matrices to get:
    [(333/6350) (644/6350)
    (-644/6350) (333/6350)]

    I did similar for the second part, but I'll spare all the numbers, since I'm messing something up....

    Further, how would I go about getting the radians? I know the formula for counterclockwise rotation, but wouldn't know how to come up with the radians of such a number...
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 25, 2008 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I don't recognize your formula for the reflection matrix so what I would do is this.
    <3, 4> is a vector in the direction of the line -4x+ 3y= 0 and <-4, 3> is a vector perpendicular to it. The reflection in that line maps <3, 4> into itself and <-4, 3> into its negative, <4, -3> Setting up the two equations
    [tex]\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]\left[\begin{array}{c}3 \\ 4\end{array}\right]= \left[\begin{array}{c}3 \\ 4\end{array}\right][/tex]
    and
    [tex]\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]\left[\begin{array}{c}-4 \\ 3\end{array}\right]= \left[\begin{array}{c}4 \\ -3\end{array}\right][/tex]
    gives 4 equations for a, b, c, d. I get
    [tex]A= \left[\begin{array}{cc}\frac{-7}{25} & \frac{24}{25} \\ \frac{14}{25} & \frac{7}{25}\end{array}\right][/tex]
    for the first reflection.

    You can do the same for the second reflection and, of course, their composition is the product of the matrices.

    I don't believe
    [(333/6350) (644/6350)
    (-644/6350) (333/6350)]

    is correct because its determinant is not 1, which must be true for a rotation matrix.
     
  4. Nov 25, 2008 #3
    Hi,

    Thanks again HallsofIvy.

    I used the unit vectors in my formula, and it seems to come out with the same answer; except I tried the technique you gave and I still come up with
    [(-7/25) (24/25)
    (24/25) (7/25)]
    for the second Matrix.

    I guess I'm making some calculation error, as
    [(333/6350) (644/6350)
    (-644/6350) (333/6350)]
    Is the matrix I get from the product AB...

    Thanks!
     
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